48-th Mathematical Olympiad in Poland Second Round, February 21–22, 1997

First Day

1. For any real number a determine the number of the ordered triples (x, y, z) of real numbers satisfying the following system of equations

\{ \begin{matrix}x + y^{2} + z^{2} & = & a \\ x^{2} + y + z^{2} & = & a \\ x^{2} + y^{2} + z & = & a \end{matrix}

2. Point P lies inside triangle ABC and satisfies the conditions:

P̸ BA = ^{}P̸ CA = \frac{1}{3}(^{}A̸BC +^{}A̸CB) .

Prove that

\frac{AC}{AB + P C} = \frac{AB}{AC + P B} .

3. Given is a set of n points (n ≥ 2); no three of the points are collinear. We colour all the line segments with endpoints in this set so that two segments with a common endpoint are of different colours. Determine the least number of colours, for which there exists such a colouring.

Second Day

4. Find all triples of positive integers with the following property: The product of any two of these numbers gives the remainder 1 upon division by the third number.

5. We have thrown k white dice and m black dice. Find the probability that the remainder upon division by 7 of the number on the faces of the white dice is equal to the remainder upon division by 7 of the number on the faces of the black dice.

6. In a cube of the edge 1, eight points are given. Prove that two of the points are the endpoints of a segment of length not greater than 1.